Principal Components in the Nonnormal Case: The Test for Sphericity.
Abstract
The limiting distribution of the likelihood ratio statistic W sub q for testing the hypothesis of equality of q characteristic roots of a covariance matrix for normal populations is studied for nonnormal populations. It is shown, both theoretically and empirically, that the limiting distribution of W sub q is not robust to departures from normality characterized by nonzero fourth cumulants and that W sub q cannot be used for these nonnormal populations. For the class of spherically symmetric populations, it is shown that the limiting distribution of W sub q is proportional to a chi-square under the null hypothesis of equality of q population roots and to a noncentral chi-square under an appropriate sequence of alternative hypotheses. A corrected test statistic, W sub q, whose limiting distribution is chi-square, is proposed to test the hypothesis of sphericity for the general class of spherically symmetric populations. Results of a Monte Carlo experiment conducted to compare the performances of W sub q and W sub q are presented for various contaminated normal models at various sample sizes. It is demonstrated that there is little difference between W sub q and W sub q for normal populations, but that dramatic improvements are gained by using W sub q for the nonnormal populations. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Oct 01, 1977
- Accession Number
- ADA049622
Entities
People
- Christine M. Waternaux
Organizations
- Stanford University